1. 4-1 Graphing Polynomial Functions

2. Polynomials
A monomial is a number, a variable, or the product of a number and one or more variables with whole number exponents. 
Monomial: 5, 2x, xyh, x2
Not a monomial: : x+2, x/y, x3/2
A polynomial is a monomial or a sum of monomials

3. Polynomial Functions A polynomial function is a function of the form
where , the exponents are all whole numbers, and the coefficients are all real numbers.
In standard form, a polynomial function is written in order of descending powers.

4. Polynomial Graphs
quadratic
cubic
quartic

5. Examples of polynomials
Degree Name Example
constant
linear
quadratic
cubic
quartic
quintic

6. Polynomials versus not polynomials
Is each function a Polynomial Function ?
If yes, what is its degree?
Function Polynomial Degree
Yes 1
Yes 2 No
Yes No

7. Evaluating Polynomial Functions
Evaluate when
Substitute 3 for x in original eqn.
Evaluate powers and multiply.
Simplify.

8. End Behavior
The end behavior of a function's graph is the behavior of the graph as x approaches positive infinity (+∞) or negative infinity (-∞).
For the graph of a polynomial function, the end behavior is determined by the function's degree and the sign of the leading coefficient.

9. End Behavior of Polynomial Functions
Degree: odd
Leading coefficient: positive
Degree: odd
Leading coefficient: negative

10. End Behavior of Polynomial Functions
Degree: even
Leading coefficient: positive
Degree: even
Leading coefficient: negative

11. End Behavior of Polynomial Functions
Example: evaluate the end behavior of the function It has a degree of 4 (even) and a leading coefficient of -0.2 (negative).
f(x) approaches -∞ as x approaches -∞
and as x approaches +∞.

12. Graphing Polynomial Functions
To graph a polynomial function, first plot points to determine the shape of the graph’s middle portion. Then connect the points with a smooth continuous curve and use what you know about end behavior to sketch the graph. Ex: graph X -2 -1 1 2
F(X) 3 -4 -3
The degree is odd and the leading coefficient is negative.
So, f(x) approaches +∞ as x approaches - ∞ and f(x) approaches - ∞ as x approaches + ∞.

13. Solving a Real Life Problem
The estimated number V (in thousands) of electric vehicles in use in the United States can be modeled by the polynomial function
with t representing time and t=1 corresponding to 2001.
a. Use a graphing calculator to graph the function for the interval 1 ≤ t ≤ 10. 

14. Solving a Real Life Problem
The estimated number V (in thousands) of electric vehicles in use in the United States can be modeled by the polynomial function
with t representing time and t=1 corresponding to 2001.
b. What was the average rate of change in the number of electric vehicles in use from 2001 to 2010?
(1, 18.57) , (10, ) where t = 1 and t = 10
The average rate of change in the number of electric vehicles in use from 2001 to 2010 is about 9.06 thousand electric vehicles per year.

15. Solving a Real Life Problem
The estimated number V (in thousands) of electric vehicles in use in the United States can be modeled by the polynomial function
with t representing time and t=1 corresponding to 2001.
c. Do you think this model can be used for years before 2001 or after 2010? Explain your reasoning.
Because the degree is odd and the leading coefficient is positive, V(t) approaches +∞ as t approaches +∞ and V(t) approaches -∞ as t approaches -∞ .
The end behavior indicates that the model has unlimited growth as t increases. While the model may be valid for a few years after 2010, in the long run unlimited growth is not reasonable.
Notice that in 2000 that v(0) = Because negative values of V(t) do not make sense given the context (electric vehicles in use), the model should not be used for years before 2001